Mathematical Methods for Physicists: A concise introduction - Site Map

THE VECTOR (CROSS OR OUTER) PRODUCT
Figure 1.9. The torque of a force about a point O.
where " ijk is de®ned by
8
< ‡1 if …i; j; k† is an even permutation of …1; 2; 3†;
" ijk ˆ 1 if …i; j; k† is an odd permutation of …1; 2; 3†;
:
0 otherwise …for example; if 2 or more indices are equal†:
…1:18†
It follows immediately that
" ijk ˆ " kij ˆ " jki ˆ" jik ˆ" kji ˆ" ikj :
There is a very useful identity relating the " ijk and the Kronecker delta symbol:
X 3
kˆ1
" mnk " ijk ˆ mi nj mj ni ; …1:19†
X
" mjk " njk ˆ 2 mn ;
j;k
X
" 2 ijk ˆ 6: …1:19a†
i;j;k
Using permutation symbols, we can now write the vector product A B as
!
!
A B ˆ
X3
A i^e i
X3
B j^e j ˆ X3 X 3
A i B j ^e i ^e j ˆ A i B j " ijk
^ek :
iˆ1
jˆ1
Thus the kth component of A B is
…A B† k ˆ X
A i B j " ijk ˆ X
" kij A i B j :
i;j
If k ˆ 1, we obtain the usual geometrical result:
…A B† 1 ˆ X
" 1ij A i B j ˆ " 123 A 2 B 3 ‡ " 132 A 3 B 2 ˆ A 2 B 3 A 3 B 2 :
i;j
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i;j
i;j
i;j;k